For single-particle items of Hamiltonian (REF ), the
time-reversal, particle-hole, and chiral symmetries exist, then the
topological insulator belongs to the symmetry class BDI of the
Altland-Zirnbauer classification and is characterized by a \(\mathbb {Z}\) 
invariant [1]}, [2]}, [3]}, [4]}.
When \(t_{1}=t_{3}\)  and \(t_{2}=t_{4}\) , the SSH4 model reduces to the common SSH model.
The SSH4 model has four bands with more mid-gap states located inside the three gaps.
However, only the zero-energy state is protected by the chiral symmetry and associated with the band-topology.
Thus, the winding number is defined to identify the property of the two negative (or positive) energy bands, which all contribute to its value.
The winding number \(w=1\)  when \(|t_{1}t_{3}|<|t_{2}t_{4}|\) , and \(w=0\)  when \(|t_{1}t_{3}|>|t_{2}t_{4}|\) .
In the presence of weak interaction \(V\) , the topological invariant become \(\mathbb {Z}_{4}\)  [5]}.
In the following, we focus on the two negative topology bands with zero-energy states, which corresponds to the half-filling occupation, i.e., \(N/L=0.5\) , with the atom number \(N\)  and lattice length \(L\) .
